Étale tame vanishing cycles over $[{\mathbb{A}}_S^1\mathrm{/}{\mathbb{G}}_{m,S}]$

We develop a theory of tame vanishing cycles for schemes over $[{\mathbb{A}}_S^1\mathrm{/}{\mathbb{G}}_{m,S}]$ in the context of étale sheaves. We show some desired properties of this formalism, among which: a compatibility with tame vanishing cycles over a (strctly) henselian trait, a compatibility with the theory of tame vanishing cycles over ${\mathbb{A}}_S^1$, a compatibility with tensor product and with duality. In the last section, we prove that monodromy-invariant vanishing cycles, introduced by the second named author, are the homotopy fixed points with respect to a canonical continuous action of ${\mu }_{\mathrm{\infty }}$ of tame vanishing cycles over $[{\mathbb{A}}_S^1\mathrm{/}{\mathbb{G}}_{m,S}]$.